Discrete Mathematics Introduction

What is
Discrete Mathematics?
Why should we teach
Discrete Mathematics?
Fitting
Discrete Mathematics
into the Curriculum
Discrete Mathematics
in Grades 7&8
Conclusion

WHAT IS DISCRETE MATHEMATICS?
Specifically, what content does discrete mathematics embody? What are some meaningful applications to use at the secondary school level? The difficulty is that discrete mathematics is an inclusive term; it embraces several topical areas of mathematics, some of which go back to early stages of mathematical development while others are more recent additions to the discipline.

Discrete mathematics includes sets, functions and relations, matrix algebra, combinatorics and finite probability, graph theory, finite differences and recurrence relations, logic, mathematical induction, and algorithmic thinking. Other topics often considered part of discrete mathematics are Boolean algebra, the mathematics of social choice, linear programming, and number theory. Because of this diversity of topics, it is perhaps preferable to view discrete mathematics simply as the mathematics that is necessary for decision making in noncontinuous situations.

Referenced from:
Implementing the Discrete Mathematics Standards: Focusing on Recursion: by Margaret J. Kenney and Stanley J. Bezuszka [NCTM Mathematics Teacher] Volume 86, number 8, November 1993 (p. 676-680).


WHY SHOULD WE TEACH DISCRETE MATHEMATICS?
Discrete mathematics has evolved as the computer has evolved. Discrete mathematics is the kind of mathematics one needs to know to communicate with a computer as designer, programmer, or user. It is necessary for all students, regardless of their choice of career path, to receive some instruction in discrete mathematics so that they will be able to function as informed citizens of an increasingly technological society.

In fact, discrete mathematics affords many students a new opportunity to experience success and enjoyment in mathematics classes. Those who have encountered numerous difficulties with computation and the complexities of mathematics in the past can be reached with appealing problems from discrete mathematics that have few formal skills as requisites. Other students who have been discouraged by the routine aspects of learning mathematics can become excited and challenged by the many intriguing problems that are typical of discrete mathematics. Discrete mathematics can be used to illustrate and emphasize effectively NCTM's four overall curriculum standards for all students. That is, discrete mathematics problems- require that many problem-solving strategies be applied to interesting real-world applications; lend themselves well to situations in which students collaborate and develop verbal and written skills in the process of solving the problem; demand the sustained use of critical thinking and reasoning procedures in working toward a solution; and promote making mathematical connections within and across disciplines through a wide range of problem types. In addition, technology is typically used to gather, process, or analyze the data integral to the problem.

Referenced from: Implementing the Discrete Mathematics Standards: Focusing on Recursion: by Margaret J. Kenney and Stanley J. Bezuszka [NCTM Mathematics Teacher . Volume 86, number 8, November 1993 (p. 676-680).


FITTING DISCRETE MATHEMATICS INTO THE CURRICULUM:
Discrete mathematics can fit into many places in the curriculum, and several possible strategies can be recommended for its implementation.
  1. Emphasize discrete mathematics topics that are already part of the curriculum. For example, matrices, counting, induction, sequences, sets, and logic are discrete-mathematics topics that are already in the curriculum.
  2. Take a "discrete" approach to old topics. For example, use matrices to solve systems of linear equations and to represent geometric transformations, or represent relations using graphs and matrices, or use recursive formulas for sequences.
  3. Teach short (two- to ten-day) units on "new'" discrete-mathematics topics. Materials are rapidly becoming available for topics like graph theory, difference equations, game theory, and linear programming. But do we have room? Yes! Many teachers are already teaching such units without eliminating other topics by teaching the units during "slack" times, such as just before and after vacation breaks. Also, one can make room in the curriculum by reducing the time spent on such topics as factoring and two column proofs, as recommended in the Standards.
  4. Teach a full-semester course on discrete mathematics. Numerous high schools around the country are already starting to take this approach. At present such a course is most commonly viewed as a senior mathematics course for better students. But a discrete-mathematics course could fit into the curriculum. In particular, discrete mathematics topics are well suited to the "core curriculum" recommendation of the Standards (see Hirsch and Schoen [1989]). That is, the same topics can be taught at different levels and with different applications. Thus, a discrete-mathematics course could be appropriate for students who now take "general math," for students who have completed first-year algebra or geometry but are not quite ready for second-year algebra, or for students who need a course to follow or replace calculus.
  5. Weave in discrete mathematics. Many discrete mathematics topics can be woven into the existing curriculum, as discussed in categories 1, 2, and 3 and also by using discrete-mathematics topics as examples and applications within the existing curriculum. For instance, a game-theory example, like finding the optimum percentage of fast balls that a pitcher should throw in the game of baseball, can be presented as an application of solving systems of linear equations or an apportionment example, like apportioning seats in a state legislature, can be presented as an application within a unit on fractions. A good resource that discusses these examples and also offers instructive and motivating videotapes is Garfunkel (1988). Thus, many strategies can be found for fitting discrete mathematics into the secondary school curriculum.

Garfunkel, Solomon. For All Practical Purposes: Introduction to Contemporary Mathematics. New York: W. H. Freeman & Co., 1988.

Hirsch, Christian R., and Harold L. Schoen. "A Core Curriculum for Grades 9-12." Mathematics Teacher 82 (December 1989):696-701.



DISCRETE MATHEMATICS IN GRADES 7 AND 8:
Although the Standards does not contain a specific recommendation concerning discrete mathematics in grades 7 and 8, some discrete-mathematics topics can easily be taught in these grades, and doing so will address many of the middle school standards. For example, the global standards of problem solving, communications, reasoning, and connections can all be addressed by teaching discrete mathematics, as can the standards on number and number relationships, patterns and functions, algebra, and geometry. We shall give examples of three discrete-mathematics topics appropriate for middle school students.


CONCLUSION:
Discrete mathematics is vital, exciting, and useful mathematics that should be taught in grades 7-12. Many topics can be taught, and they can be fit into the curriculum in a variety of ways. Some teacher retraining is necessary, but the small investment of time and effort needed to begin teaching discrete mathematics has substantial payoffs in terms of a richer curriculum and better-prepared students.

Abstract and basic descriptive record provided by the ERIC Clearinghouse system.

[Permission from all of the above authors is pending...]


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July 16, 1996